VariationalMethods`
VariationalMethods`

# NVariationalBound

NVariationalBound[f,u[x],{x,xmin,xmax},ut,{a,a0},{b,b0},]

numerically searches for values of the parameters a, b, ... of a trial function ut, starting from a=a0, b=b0, ..., that extremize the functional , where the integrand f is a function of u, its derivatives, and x.

NVariationalBound[f,u[x,y,],{{x,xmin,xmax},},ut,{a,a0},{b,b0},]

searches for values of the parameters of a trial function of two or more variables.

NVariationalBound[{f,g},u[x],{x,xmin,xmax},ut,{a,a0},{b,b0},]

searches for values of the parameters that extremize the ratio , where the integrands f and g are functions of u, its derivatives, and x.

# Details

• To use NVariationalBound, you first need to load the Variational Methods Package using Needs["VariationalMethods`"].
• NVariationalBound returns the extremal value of the functional as well as the optimal parameter values.
• NVariationalBound uses FindMinimum to search for values of the parameters that extremize the functional.
• A parameter specification of {a,a0,a1} searches for an extremum using a0 and a1 as the first two values of a, avoiding the use of derivatives.
• A parameter specification of {a,a0,amin,amax} searches for an extremum, stopping the search if a ever gets outside the range amin to amax.

# Examples

## Basic Examples(1)

Eigenvalue problem for a third-order ordinary differential equation:

The solution fits the equation well in this case: